Differences between revisions 6 and 13 (spanning 7 versions)

Derivative

A derivative is an instantaneous rate of change with respect to an input variable. It is a ratio of differentials.

Contents

Derivative
1. Rules

Rules

The basic rules/identities are:

Rule	Formulation	Defined for...
constants
constant factors
polynomials (power rule)
exponentiation
exponentiation (generalized)		a > 0
logarithms		x > 0
logarithms (generalized)		x > 0 and a > 0

For trigonometric functions:

Rule	Formulation	Defined for...
sine
cosine
tangent
inverse sine		-1 < x < 1
inverse cosine		-1 < x < 1
inverse tangent

Chain Rule

For composite functions like e^2x or sin(2x), the chain rule must be applied. Let f be the entire function as-is (e.g., e^2x), h be the 'inner function (e.g., 2x), and g be the 'outer' function (e.g., e^h(x)).

An inconvenient function h(x) can be rewritten as e^ln(h(x)), which can be evaluated using this rule.

Product Rule

Consider a function like f(x) = g(x)h(x). Evaluate as:

This product rule holds for vector multiplication. That is, given a vector f that was defined as a dot product like f = g ⋅ h, the derivative (i.e., over time) of f can be calculated from known derivatives of the components. In this case, try:

The same is true of an f defined as a cross product like f = g × h. Try:

Quotient Rule

Consider a function like f(x) = g(x)/h(x). Evaluate as:

Properties

Derivatives are linear: given a function defined like f(x) = αg(x) + βh(x):

The follows from the total differential; substitute g and h for g(x) and h(x):

f = gh

df = f_gdg + f_hdh

df/dx = f_g(dg/dx) + f_h(dh/dx)

And clearly the partial derivatives f_g and f_h are equal to h and g respectively, giving:

df/dx = h(dg/dx) + g(dh/dx)

Substituting back in the original functions gives the product rule.

CategoryRicottone

-  ⇤ ← Revision 6 as of 2025-11-04 01:16:45 → 
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+   ← Revision 13 as of 2026-06-18 12:08:47 → ⇥
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-||'''Rule'''                  ||'''Formulation'''            ||'''Defined for...'''   ||
||constants                   ||{{attachment:const.svg}}     || ||
||constant factors            ||{{attachment:constfact.svg}} || ||
||polynomials                 ||{{attachment:polynomial.svg}}|| ||
||exponentiation              ||{{attachment:e.svg}}         || ||
||exponentiation (generalized)||{{attachment:exp.svg}}       ||''a > 0''              ||
||logarithms                  ||{{attachment:ln.svg}}        ||''x > 0''              ||
||logarithms (generalized)    ||{{attachment:log.svg}}       ||''x > 0'' and ''a > 0''||
+||'''Rule'''                    ||'''Formulation'''                                  ||'''Defined for...'''   ||
||constants                     ||{{attachment:const.svg}}                           || ||
||constant factors              ||{{attachment:constfact.svg}}                       || ||
||polynomials ('''power rule''')||{{attachment:polynomial.svg}}                      || ||
||exponentiation                ||{{attachment:e.svg}}                               || ||
||exponentiation (generalized)  ||{{attachment:exp.svg}}                             ||''a > 0''              ||
||logarithms                    ||{{attachment:ln.svg}}                              ||''x > 0''              ||
||logarithms (generalized)      ||{{attachment:log.svg}}                             ||''x > 0'' and ''a > 0''||
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+=== Chain Rule ===

For composite functions like ''e^2x^'' or ''sin(2x)'', the '''chain rule''' must be applied. Let ''f'' be the entire function as-is (e.g., ''e^2x^''), ''h'' be the 'inner function (e.g., ''2x''), and ''g'' be the 'outer' function (e.g., ''e^h(x)^'').

{{attachment:chain.svg}}

An inconvenient function ''h(x)'' can be rewritten as ''e^ln(h(x))^'', which can be evaluated using this rule.



=== Product Rule ===

Consider a function like ''f(x) = g(x)h(x)''. Evaluate as:

{{attachment:prod1.svg}}

This '''product rule''' holds for vector multiplication. That is, given a vector ''f'' that was defined as a [[Calculus/VectorOperations#Dot_Product|dot product]] like ''f = g ⋅ h'', the derivative (i.e., [[Calculus/SpeedVelocityAndAcceleration|over time]]) of ''f'' can be calculated from known derivatives of the components. In this case, try:

{{attachment:prod2.svg}}

The same is true of an ''f'' defined as a [[Calculus/VectorOperations#Cross_Product|cross product]] like ''f = g × h''. Try:

{{attachment:prod3.svg}}



=== Quotient Rule ===

Consider a function like ''f(x) = g(x)/h(x)''. Evaluate as:

{{attachment:quot.svg}}
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-Derivatives are linear: given a function defined like ''f(x) = αg(x) + βh(x)'', {{attachment:sum.svg}}.
+Derivatives are linear: given a function defined like ''f(x) = αg(x) + βh(x)'':
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+Line 74:
-The '''product rule''' states that, given a function defined like ''f(x) = g(x)h(x)'', {{attachment:prod.svg}}. This follows from the [[Calculus/Differential|total differential]]; substitute ''g'' and ''h'' for ''g(x)'' and ''h(x)'':
+{{attachment:sum.svg}}.

The follows from the [[Calculus/Differential|total differential]]; substitute ''g'' and ''h'' for ''g(x)'' and ''h(x)'':
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-The product rule holds for vector multiplication; that is, for a [[Calculus/VectorOperations#Dot_Product|dot product]]:

''f = g ⋅ h''

''df/dx = h ⋅ (dg/dx) + g ⋅ (dh/dx)''

...and also for a [[Calculus/VectorOperations#Cross_Product|cross product]]:

''f = g × h''

''df/dx = h × (dg/dx) + g × (dh/dx)''

Diff for "Calculus/Derivative"